Faddeev-Jackiw Approach to Classical Constrained Systems

TL;DR

This work adapts the Faddeev-Jackiw symplectic approach to quantize several classical constrained systems with singular Lagrangians, analyzing constraint structures, gauge symmetries, and the role of Lagrange multipliers. By rewriting each system in first-order form and iteratively constructing the symplectic two-form, the authors extract constraints from zero-modes, fix gauges when needed, and obtain non-singular inverse matrices whose elements yield the basic brackets. The resulting brackets align with Dirac brackets, while the FJ framework also reveals multipliers' physical roles within the extended bracket structure. A MATLAB-oriented outline is provided to automate the symplectic formulation and bracket extraction, underscoring the method’s practicality as an alternative to Dirac-Bergmann for constrained dynamics.

Abstract

We accomplish the quantization of a few classical constrained systems à la (modified) Faddeev-Jackiw formalism. We analyze the constraint structure and obtain basic brackets of the theory. In addition, we disclose the gauge symmetries within the symplectic framework. We also provide an interpretation for Lagrange multipliers and outline a MATLAB implementation algorithm for symplectic formulation.